An improvement on Waring-Goldbach problem for unlike powers. (English) Zbl 1274.11154

Liu, Zhixin

Acta Math. Hung. 130, No. 1-2, 118-139 (2011).

Summary: Let \(p_i\) be prime numbers. In this paper, it is proved that for any integer \(k\geqq 5\), with at most \(O\big(N^{1-\frac{1}{3k\times 2^{k-2}}+\varepsilon}\big)\) exceptions, all positive even integers up to \(N\) can be expressed in the form \(p^2_2+ p^3_3+p^5_5+p^k_k\). This improves the result \(O\big(\frac{N}{\log^c N}\big)\) for some \(c > 0\) due to M.-G. Lu and Z. Shan [Kexue Tongbao, Foreign. Lang. Ed. 27, 246–250 (1982; Zbl 0497.10037); J. China Univ. Sci. Tech., Suppl. I, 1–8 (Chinese)(1982)], and it is a generalization for a series of results of X.-M. Ren and K.-M. Tsang [Acta Math. Sin., Engl. Ser. 23, No. 2, 265–280 (2007; Zbl 1128.11043), Acta Math. Sin., Chin. Ser. 50, No. 1, 175–182 (2007; Zbl 1121.11312)], and C. Bauer [Rocky Mt. J. Math. 38, No. 4, 1073–1090 (2008; Zbl 1232.11101), a.o.] for the problem in the form \(p^2_2 + p^3_3+p^4_4 +p^5_5\). This method can also be used for some other similar forms.

MSC:

11P32	Goldbach-type theorems; other additive questions involving primes
11P05	Waring’s problem and variants
11P55	Applications of the Hardy-Littlewood method

Keywords:

Waring-Goldbach problem; circle method; exceptional set

Citations:

Zbl 0497.10037; Zbl 1128.11043; Zbl 1121.11312; Zbl 1232.11101

References:

[1]	C. Bauer, On a problem of the Goldbach–Waring type, Acta Math. Sinica, New Series, 14 (1998), 223–234. · Zbl 0901.11030
[2]	C. Bauer, An improvement on a theorem of the Goldbach–Waring type, Rocky Mountain J. Math., 31 (2001), 1151–1170. · Zbl 1035.11047
[3]	C. Bauer, A remark on a theorem of the Goldbach Waring type, Studia Sci. Math. Hungar., 41 (2004), 309–324. · Zbl 1064.11065
[4]	C. Bauer, A Goldbach–Waring problem for unequal powers of primes, Rocky Mountain J. Math., 38 (2008), 1073–1090. · Zbl 1232.11101
[5]	S. Choi and A. Kumchev, Mean values of Drichlet polynomials and application to linear equations with prime variables, Acta Arith., 123 (2006), 125–142. · Zbl 1182.11048
[6]	R. Cook, On sums of powers of integers, J. Number Theory, 4 (1979), 516–528. · Zbl 0408.10034
[7]	P. X. Gallagher, A large sieve density estimate near {\(\sigma\)}=1, Invent. Math., 11 (1970), 329–339. · Zbl 0219.10048
[8]	L. K. Hua, Additive Theory of Prime Numbers, Science Press (Beijing, 1957); English Version, Amer. Math. Soc. (Rhode Island, 1965).
[9]	N. M. Huxley, Large values of Dirichlet polynomials III, Acta Arith., 26 (1975), 435–444. · Zbl 0268.10026
[10]	J. Y. Liu and M. C. Liu, The exceptional set in the four prime squares problem, Illinois J. Math., 44 (2000), pp. 272–293. · Zbl 0942.11044
[11]	A. Kumchev, On Weyl sums over primes and almost primes, Michigan Math. J., 54 (2006), 243–268. · Zbl 1137.11054
[12]	M. G. Lu and Z. Shan, A problem of Waring–Goldbach type, J. China Univ. Sci. Tech., Suppl. I (1982), 1–8 (in Chinese). · Zbl 0497.10037
[13]	K. Prachar, Über ein Problem vom Waring–Goldbach’schen Typ, Monatsh. Math., 57 (1953), 66–74. · Zbl 0050.04003
[14]	K. Prachar, Primzahlverteilung, Springer Verlag (Berlin, 1978).
[15]	X. M. Ren and Kai-Man Tsang, Waring–Goldbach problem for unlike powers, Acta Math. Sinica, English Series (2), 23 (2007), 265–280. · Zbl 1128.11043
[16]	X. M. Ren and Kai-Man Tsang, Waring–Goldbach problem for unlike powers (II), Acta Math. Sinica (Chinese Series), 50 (2007), 175–182. · Zbl 1121.11312
[17]	K. F. Roth, A problem in additive number theory, Proc. London Math. Soc., 53 (1951), 381–395. · Zbl 0044.03601
[18]	K. Thanigasalam, On certain sequences of integers, Trans. Amer. Math. Soc., 200 (1974), 199–205. · Zbl 0292.10043
[19]	E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd edn., Clarendon Press (Oxford, 1986). · Zbl 0601.10026
[20]	I. M. Vinogradov, Elements of Number Theory, Dover Publications (New York, 1954). · Zbl 0057.28201